Emmy Noether

Lived 1882 – 1935.

Emmy Noether is probably the greatest female mathematician who has ever lived. She transformed our understanding of the universe with Noether’s theorem and then transformed mathematics with her founding work in abstract algebra.

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Beginnings

Amalie Emmy Noether was born in the small university city of Erlangen in Germany on March 23, 1882. Her father, Max Noether, was an eminent professor of mathematics at the University of Erlangen. Her mother was Ida Amalia Kaufmann, whose family were wealthy wholesalers.

Young Emmy was brought up as a typical girl of her era: helping with cooking and running the house – she admitted later that she had little aptitude for these sorts of things. Her mother was a skilled pianist, but Emmy did not enjoy piano lessons. Her main passion was dancing.

She also loved mathematics, but she knew that the rules of the time meant she would not be allowed to follow in her father’s footsteps to become a university academic.

After completing high school – she attended the Municipal School for Higher Education of Daughters in Erlangen – she trained to become a school teacher, qualifying in 1900, aged 18, to teach English and French in girls’ schools.

Although a career in teaching offered her financial security, her love of mathematics proved to be too strong. She decided to abandon teaching and apply to the University of Erlangen to observe mathematics lectures there. She could only observe lectures, because women were not permitted to enroll officially at the university.

Between 1900 and 1902 Emmy studied mathematics at Erlangen. In July 1903 she traveled to the city of Nürnberg and passed the matriculation examination allowing her to study mathematics (but not officially enroll) at any German university.

At the Center of the Universe for a semester

Emmy chose to go for a semester to the University of Göttingen, then home to the most prestigious school of mathematics in the world.

Many of the greatest mathematicians in history had taught and been taught at Göttingen, including Carl Friedrich Gauss and Bernhard Riemann. Emmy attended lectures given by:

Hermann Minkowski, the esteemed mathematician who had taught Albert Einstein, and

David Hilbert, probably the twentieth century’s most outstanding mathematician

Doctorate in Mathematics

In 1904 Emmy was overjoyed to learn that her hometown university, Erlangen, had decided women should be permitted full access.

Professor Paul Gordan. Emmy Noether was the only student he ever accepted as a Ph.D. candidate.

She was accepted as a Ph.D. student by the renowned mathematician Paul Gordan. Gordan was 67 when Emmy started work with him. She was the only student he ever accepted as a Ph.D. candidate.

Gordan was known among mathematicians as “the king of invariant theory.” Emmy made exceptional progress in this field, which would later lead to her making her remarkable discovery in physics

Dr. Noether, Mathematics Lecturer

In 1908 Noether was appointed to the position of mathematics lecturer at Erlangen. Unfortunately, it was an unpaid position. This was not especially unusual in Germany for a first lecturing job. The great chemist Robert Bunsen’s first lecturing position was without pay at the University of Göttingen.

Noether’s parents supported her as much as they could through this time, her father recognizing something rather special in his daughter’s capabilities. Nevertheless, her life was a struggle financially.

While working as a lecturer, Noether became fascinated by work David Hilbert had done in Göttingen. The work was more abstract than any she had done at Erlangen. She began applying, stretching and modifying Hilbert’s methods. This was her first heavyweight encounter with abstract algebra, mathematical territory in which she would soon become a powerful innovator.

An Invitation from Hilbert

David Hilbert familiarized himself with Noether’s research. Like her father, he recognized there was something special there. By this stage in his career Hilbert was concerned mainly with physics, which he believed needed help from the best mathematicians, famously declaring:

Physics is actually too hard for physicists.

David Hilbert

In 1913 and 1914 Noether exchanged letters with David Hilbert and his Göttingen colleague Felix Klein discussing Einstein’s Relativity Theory.

In 1915 Hilbert invited her to become a lecturer in Göttingen. Unfortunately this provoked a storm of protest from the history and linguistics faculties who did not think it appropriate that a woman should be teaching men, particularly since Germany was at war – World War 1: 1914 – 1918. Although in general the mathematics and science faculties supported Noether, they could not overcome the opposition from the humanities faculties.

Noether was so eager to join Hilbert’s department in Göttingen that, to overcome Hilbert’s opponents, she agreed not to be formally appointed as a lecturer and to receive no pay. Her father continued supporting her financially (sadly her mother died in 1915) and the lectures she gave were advertised as lectures by Professor Hilbert, with assistance from Dr. E. Noether.

Noether’s Theorem

Hilbert, Einstein, Noether, and the General Theory of Relativity

Einstein visits Göttingen

In 1915 Albert Einstein had been struggling to formulate his General Theory of Relativity mathematically. He visited David Hilbert in Göttingen and discussed the issues. The result was that Einstein overcame his issues and published his theory before the year end. Hilbert published his own version of the theory, in a different mathematical form.

Einstein’s General Relativity Breaks the Law

Hilbert now discussed one particular problem with Noether. He was deeply concerned that, despite its attractions, Relativity Theory was breaking one of the ‘unbreakable’ conservation laws of physics. He believed that her expertise in invariant theory could be helpful.

Certain quantities in physics may not be created or destroyed, such as energy. Energy can change its form – such as kinetic to thermal – but the total energy stays constant – energy is said to be conserved.

In General Relativity Theory however, there was a problem: it was possible for an object which lost energy by emitting gravity waves to speed up. An object with less energy should slow down, not speed up! It seemed that the conservation of energy law was being broken.

A Problem Archimedes Would Have Loved

In the end, the problem was one of symmetry. Over 2000 years earlier the greatest mathematician of antiquity – perhaps ever – had been buried with a carving of a sphere within a cylinder on his tomb. This was Archimedes, who believed his greatest achievement had been discovering and proving the formula for the volume of a sphere.

A perfect sphere is highly symmetrical. Whichever way you rotate it, and from whichever angle you view it, it always looks the same. A cylinder, on the other hand, is less symmetrical; but there is still some symmetry. If you turn it upside down, for example, it looks the same.

Physicists need to use equations whose symmetry is as sphere-like as possible. They last thing they want is equations that change depending on where you are viewing the universe from. In physics jargon we say that we need the laws of the universe to be space invariant. We don’t want them to look different in one city from another or in one galaxy from another.

We also need these laws to be time invariant. We don’t want the laws of physics in an hour’s time to be different from the laws right now.

Noether to the Rescue

Noether hit the ground running in Göttingen. In the year she arrived she proved something remarkable – something so beautiful and profound that it changed the face of physics forever – Noether’s Theorem, which she eventually published in 1918.

Her famous theorem was born when Noether considered Hilbert and Einstein’s problem: that General Relativity Theory seemed to break the law of conservation of energy.

Noether discovered that for every invariant (i.e. symmetry) in the universe there is a conservation law. Equally, for every conservation law in physics, there is an invariant. This is called Noether’s Theorem and it describes a fundamental property of our universe.

For example, Noether’s Theorem shows that the law of conservation of energy is actually a consequence of time invariance in classical physics. Or alternatively that time invariance is caused by the law of conservation of energy.

Another example is that the law of conservation of electric charge is a consequence of the global gauge invariance of the electromagnetic field. And vice versa.

Opening up Strange New Worlds

With Noether’s Theorem, physicists now owned a very powerful new concept. They could propose abstract symmetries, knowing there must be a conservation law attached to each of them. They could then figure out the conservation law.

Noether’s Theorem has the power to answer questions others cannot – particularly in particle physics. It is important on two levels:

it allows practical calculations to be made, and

when physicists theorize about any new system they can imagine, Noether’s theorem allows them to gain an insight into the properties of that system and determine if that system is possible or should be discarded.

Einstein’s Problem Solved

Noether’s Theorem also solved the worrying puzzle in General Relativity that Emmy Noether had initially set out to solve. Her theorem shows that if matter and gravity are considered to be one unified quantity rather than separate quantities, then there is no violation of any conservation law.

“Yesterday I received a very interesting paper on invariants from Miss Noether. I’m impressed that these things can be seen in such a general way. It would do the old guard at Göttingen no harm to be sent back to school under Miss Noether. She knows her stuff.”

Albert Einstein, 1879 to 1955

1918

Einstein became vocal about Göttingen’s refusal to appoint Noether as a lecturer, telling Felix Klein:

“After receiving Miss Noether’s new paper, I once again feel that depriving her of a teaching job is a great injustice. I would like vigorous steps to be taken with the Ministry. If you do not think this is possible, then I will go the trouble of doing it myself.”

Albert Einstein, 1879 to 1955

1918

At last: some career progress

With the end of World War 1, in which so many men had died or been badly injured, came a change in German society. It became acceptable for women to work in occupations previously reserved for men. Combined with Noether publishing her brilliant theorem, her academic progress could no longer be blocked.

At the age of 37 she became a tenured lecturer at Göttingen. However, she still received no pay from a now war-impoverished Germany. Her father died when she was 39, leaving her a small inheritance.

It was only when she reached the age of 40 that Noether finally began to receive a salary.

Abstract Algebra

Noether’s Theorem had revolutionized physics. In 1919 the full force of her powerful mind turned towards pure mathematics. In this discipline, she was one of the principle architects of abstract algebra. Her name is remembered in many of its concepts, structures and objects, such as:

mathematical rings – she established the modern axiomatic definition of the commutative ring and developed the basis of commutative ring theory

commutative number fields

linear transformations

noncommutative algebras – Hermann Weyl credited Noether with representations of noncommutative algebras by linear transformations, and their application to the study of commutative number fields and their arithmetics

“And one can­not read the scope of her ac­com­plish­ments from the in­di­vidu­al res­ults of her pa­pers alone: she ori­gin­ated above all a new and epoch-mak­ing style of think­ing in al­gebra.”

Hermann Weyl, 1885 to 1955

1935

“If Emmy Noether could have been at the 1950 Congress, she would have felt very proud. Her concept of algebra had become central in contemporary mathematics. And it has continued to inspire algebraists ever since.”

Garrett Birkhoff, 1911 to 1996

The Rise of Modern Algebra, 1976

Expulsion from Germany: moving to America

In the early 1930s Noether’s career was finally taking off. Her name was becoming known, and she was receiving invitations to speak at important mathematics conferences.

Then, in January 1933, everything changed. Adolf Hitler came to power. By April of that year Noether, who was Jewish, had been dismissed from the University of Göttingen by order of the Prussian Ministry for Sciences, Art, and Public Education. Sadly, in Nazi ideology Emmy Noether’s religion was of more significance than her extraordinary genius.

Fortunately, her genius was valued somewhere. Bryn Mawr College in Pennsylvania, USA – a women’s college – obtained a grant from the Rockefeller Foundation and, in October 1933, Emmy Noether sailed on the Bremen to begin work as a lecturer in America.

The following year she also began lecturing at the Institute for Advanced Study in Princeton.

A year later she was dead.

Some Personal Details and The End

Noether was totally devoted to mathematics and talked of little else. She never married and had no children. She cared little for her appearance and less for social conventions; she was not a shrinking violet – she spoke loudly and forcibly. She could be very blunt when she disagreed with anyone on a mathematical issue, and people with whom she disagreed could feel rather bruised mentally.

On the other hand, she was very kind, considerate, and unselfish with everyone, and would go out of her way to ensure her Ph.D. students got full credit for their work, even when she had contributed significantly to it herself.

Only students who were very bright and fully prepared benefited from her rather disorganized lectures – rather like Willard Gibbs‘s students.

To her advanced students, she would present ideas at the forefront of modern mathematics – concepts that she herself was currently working on. This was of great benefit to her best students, who were able to publish research papers based on new, entirely original ideas Noether had been discussing in her lectures. Her best lessons were delivered informally, in conversations, or when out walking with her students, for whom she always had time.

“In her… apartment in Göttingen a large group would get together eagerly and often. People of diverse scholarly reputations and positions — from Hilbert, Landau, Brauer and Weyl to the youngest students — would gather at her home and feel relaxed and unconstrained, as in few other scientific salons in Europe. These ‘festive evenings’ in her apartment were arranged on any possible occasion…”

Pavel Aleksandrov, 1896 to 1982

1935

Emmy Noether died in Bryn Mawr at the age of 53 on April 14, 1935. She died of complications a few days after an operation to remove a tumor from her pelvis. The cause of death was possibly a viral infection. Her ashes were buried under the cloisters of Bryn Mawr College’s M. Carey Thomas Library.

“In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians.”

Albert Einstein, 1879 to 1955

1935

“…you were a great woman mathematician – I have no reservations in calling you the greatest that history has known.”